Exercise--Related rates and implicit differentiation

1. Each edge of a cube is expanding at the rate of 1 centimeter (cm) per second. How fast is the volume changing when the length of each edge is (a) 5 cm? (b) 10 cm? (c) $x$ cm?

2. An airplane flies in level flight at constant velocity, eight miles above the ground. (In this exercise assume the earth is flat.) The flight path passes directly over a point P on the ground. The distance from the plane to P is decreasing at the rate of 4 miles per minute at the instant when this distance is 10 miles. Compute the velocity of the plane in miles per hour.

3. A baseball diamond is a 90-foot square. A ball is batted along the third-base line at a constant speed of 100 feet per second. How fast is its distance from first base changing when (a) it is halfway to third base? (b) it reaches third base?

4. A boat sails parallel to a straight beach at a constant speed of $12$ miles per hour, staying $4$ miles offshore. How fast is it approaching a lighthouse on the shoreline at the instant it is exactly $5$ miles from the lighthouse ?

5. A reservoir has the shape of a right-circular cone. The altitude is 10 feet, and the radius of the base is 4 ft. Water is poured into the reservoir at a constant rate of 5 cubic feet per minute. How fast is the water level rising when the depth of the water is 5 feet if (a) the vertex of the cone is up? (b) the vertex of the cone is down?

6. A water tank has the shape of a right-circular cone with its vertex down. Its altitude is 10 feet and the radius of the base is 15 feet. Water leaks out of the bottom at a constant rate of 1 cubic foot per second. Water is poured into the tank at a constant rate of $c$ cubic feet per second. Compute $c$ so that the water level will be rising at the rate of 4 feet per second at the instant when the water is 2 feet deep.

7. Water flows into a hemispherical tank of radius $10$ feet (flat side up). At any instant, let $h$ denote the depth of the water, measured from the bottom, $r$ the radius of the surface of the water, and $V$ the volume of the water in the tank. Compute $dV/dh$ at the instant when $h=5$ feet. If the water flows in at a constant rate of $5\sqrt{3}$ cubic feet per second, compute $dr/dt$, the rate at which $r$ is changing, at the instant $t$ when $h=5$ feet.

8. A variable right triangle $ABC$ in the $xy$-plane has its right angle at vertex $B$, a fixed vertex $A$ at the origin, and the third vertex $C$ restricted to lie on the parabola MATH The point $B$ starts at the point $(0,1)$ at time $t=0$ and moves upward along the $y$-axis at a constant velocity of $2$ cm/sec. How fast is the area of the triangle increasing when $t=7/2$ sec?

9. The radius of a right-circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as the radius. When the radius is 1 foot the altitude is 6 feet. When the radius is 6 feet, the volume is increasing at a rate of 1 cubic foot per second. When the radius is 36 feet, the volume is increasing at a rate of $n$ cubic feet per second, where n is an integer. Compute $n$.

10. A particle is constrained to move along a parabola whose equation is $y=x^2$. (a) At what point on the curve are the abscissa and the ordinate changing at the same rate? (b) Find this rate if the motion is such that at time $t$ we have $x=\sin t$ and $y=\sin ^2t$.

11. The equation $x^3+y^3=1$ defines $y$ as one or more functions of $x$. (a) Assuming the derivative $y^{\prime }$ exists, and without attempting to solve for $y$, show that $y^{\prime }$ satisfies the equation MATH. (b) Assuming the second derivative MATH exists, show that MATH whenever $y\ne 0$.

12. If $0<x<5$, the equation $x^{1/2}+y^{1/2}=5$ defines $y$ as a function of $x$. Without solving for $y$, show that the derivative $y^{\prime }$ has a fixed sign. (You may assume the existence of $y^{\prime }$.)

13. The equation $3x^2+4y^2=12$ defines $y$ implicitly as two functions of $x $ if MATH. Assuming the second derivative $y^{\prime \prime }$ exists, show that it satisfies the equation MATH.

14. The equation $x\sin xy+2x^2{}=0$ defines $y$ implicitly as a function of $x$. Assuming the derivative $y^{\prime }$ exists, show that it satisfies the equation MATH

15. If $y=x^{r}$, where $r$ is a rational number, say $r=m/n$, then $y^{n}=x^{m}$. Assuming the existence of the derivative $y^{\prime }$, derive the formula MATH using implicit differentiation and the corresponding formula for integer exponents.

L'Hôpital's Rule

The Cauchy Mean Value Theorem is the basic tool needed to prove a theorem which facilitates evaluation of limits of the form
MATH
when
MATH

Theorem. Suppose that
MATH
and and suppose also that
MATH
exists. Then
MATH
exists, and
MATH

PROOF. The hypothesis that MATH exists contains two implicit assumptions:

(1) there is an interval MATH such that $f^{\prime }(x)$ and $g^{\prime }(x)$ exist for all $x$ in MATH except, perhaps, for $x=\alpha $,

(2) in this interval MATH with, once again, the possible exception of $x=\alpha .$

On the other hand, $f$ and $g$ are not even assumed to be defined at $\alpha $. If we define MATH (changing the previous values of $f(\alpha )$ and $g(\alpha )$, if necessary), then $f$ and $g$ are continuous at $\alpha $. If MATH, then the Mean Value Theorem and the Cauchy Mean Value Theorem apply to $f$ and $g$ on the interval $[\alpha ,x]$ (and a similar statement holds for MATH). First applying the Mean Value Theorem to $g$, we see that $g(x)\ne 0$, for if $g(x)=0$ there would be some $x_{1}$ in $(\alpha ,x)$ with MATH, contradicting (2). Now applying the Cauchy Mean Value Theorem to $f$ and $g$, we see that there is a number $b$. in $(\alpha ,x)$ such that
MATH
or
MATH
Now $b$ approaches $a$ as $x$ approaches $\alpha $, because $b$, is in $(\alpha ,x)$; since MATH exists, it follows that
MATH

Uniform Continuity

Definition. The function $f$ is uniformly continuous on an interval $A$ if for every $\epsilon >0$ there is some $\delta >0$ such that, for all $x$ and $y$ in $A,$


MATH

Lemma. Let $a<b<c$ and let $f$ be continuous on the interval $[a,c].$ Let $\epsilon >0$, and suppose that statements (i) and (ii) hold:

(i) if $x$ and $y$ are in $[a,b]\;$and $\left| x-y\right| $ $<$ $\delta _1$, then MATH $<\epsilon $,

(ii) if $x$ and $y$ are in $[b,c]\;$and $\left| x-y\right| $ $<$ $\delta _2$, then MATH

Then there is a $\delta $$>$ 0 such that
MATH

Proof. Since $f$ is continuous at $b$, there is a $\delta _3$ $>$ 0 such that
MATH

It follows that

(iii) if MATH and MATH then MATH

Choose $\delta $ to be the minimum of MATH and $\delta _3$. We claim that this $\delta $ works. In fact, suppose that $x$ and $y$ are any two points in $[a,c]$ with MATH. If $x$ and $y$ are both in $[a,b],$ then MATHby (i); and if $x$ and $y$ are both in $[b,c],$ then MATHby (ii). The only other possibility is that

$x<b<y$ or $y<b<x.$

In either case, since $\left| x-y\right| $ $<\delta $ , we also have $\left| x-b\right| $ $<\delta $ and $\left| y-b\right| $ $<\delta .$ So MATHby (iii).

THEOREM If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b].$

PROOF: For $\epsilon >0$ let's say that $f\;$is $\epsilon -$good on $[a,b]$ if there is some $\delta >0$ such that, for all $y$ and $z$ in $[a,b],$if $\left| y-z\right| $ $<\delta $, then MATH. Then we're trying to prove that $f$ is $\epsilon $-good on $[a,b]$ for all $\epsilon >0.$

Consider any particular $\epsilon >0$. Let


MATH
Then $A\ne \phi \;$(since a is in $A$), and $A$ is bounded above (by $b$), so $A$ has a least upper bound $\alpha $. We really should write $\alpha _\epsilon $, since $A$ and $\alpha $ might depend on $\epsilon $. But we won't since we intend to prove that $\alpha =b$, no matter what $\epsilon \;$is.

Suppose that we had $\alpha <b$. Since $f$ is continuous at $\alpha $, there is some $\delta _0$$>0$ such that, if MATH $<\delta _0$, then MATH. Consequently, if MATH $<\delta _0$ and MATH, then MATH So $f$ is surely $\epsilon $-good on the interval MATH. On the other hand, since $\alpha $ is the least upper bound of $A$, it is also clear that $f$ is $\epsilon $-good on MATH. Then the above Lemma implies that $f$ is $\epsilon $-good on MATH, so $\alpha +\delta _0$ is in A, contradicting the fact that $\alpha $ is an upper bound.

To complete the proof we just have to show that $\alpha =b$ is actually in $A$. The argument for this is practically the same: Since $f$ is continuous at $b$, there is some $\delta _{0}>0$ such that, if MATH, then MATHSo $f$ is $\epsilon $-good on $[b-\delta _{0},b]$. But $f$ is also $\epsilon $-good on $[a,b-\delta _{0}]$, so the Lemma implies that $f$ is $\epsilon $-good on $[a,b]$.

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